We consider the extension of the basic optimal decision problem to situations involving multiple decision makers pursuing their individual distinct objectives, in competition with each other. The policy adopted by one decision maker affects the policies of others. Such competitive situations arise among individuals, firms as well as nations. A solution to the problem is the Nash equilibrium strategy which ensures that no decision maker will benefit by deviating from this equilibrium. We introduce the best replay algorithm as the intuitive solution to the computation of equilibria and discuss relaxed variants with improved convergence properties. We consider Newton-type algorithms. These aim to satisfy the optimality conditions for each player. These algorithms are quite general and applicable to a wide range of problems involving the solution of nonlinear systems of equations
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